Pi (π or Π) is an important mathematical constant, used in many science applications. The letter ”π” comes from the Greek alphabet. Three facts to know : • π is an irrationnal number, wich means its representation does not have an end. • π is the ratio of a circle’s circumference to its diameter. It is usually known as 3.141592... in decimal representation. • π is used since Antiquity
Pi (π or Π) is an important mathematical constant, used in many science applications. The letter ”π” comes from the Greek alphabet. Three facts to know : • π is an irrationnal number, wich means its representation does not have an end. • π is the ratio of a circle’s circumference to its diameter. It is usually known as 3.141592... in decimal representation. • π is used since Antiquity
Pi (π or Π) is an important mathematical constant, used in many science applications. The letter ”π” comes from the Greek alphabet. Three facts to know : • π is an irrationnal number, wich means its representation does not have an end. • π is the ratio of a circle’s circumference to its diameter. It is usually known as 3.141592... in decimal representation. • π is used since Antiquity
and beginning of calculations (the following lists are not exhaustive) ∼ 2’600 BC Egyptian, Babylonian and Indian civilisations ∼ 900 BC use π for different constructions 434 BC Archimede tried to calculate approximately : π ≈ 3.1429 130 Zhang Heng finds π ≈ √ 10 : 3.162277... ... 800 Al Khwarizmi finds π ≈ 3.1416 1200 Fibonacci finds π ≈ 3.141818 ...
π is used in some formulas, such as : • Coulomb’s law (electric force) F = |q1 q2 | 4π 0 r2 • Kepler’s third law constant (co-orbiting bodies) G(M + m) = ( 2π P )2a3 • Einstein’s Cosmological constant Λ = 8πGρvac
Geometry, π is used to calculate circumference (2πr, where r is the diameter) and area (πr2) of circles. Complexe numbers (in C) For example, a complex number z can be write like : az = r(cos φ + sin φ) And for i where i2 = −1 : eiπ = −1
Geometry Archimede tried to calculate π with Geometry. Here is how we can do : 1. Let’s make an equilateral triangle. Every side mesure 1cm. 2. Then, we make a hexagone with six triangles. We can say that our hexagone is almost a circle. 3. We calculate the perimeter of the hexagon (p = 6 ∗ 1 = 6cm) and the longest diagonal (d = 2 ∗ 1 = 2cm). Then, we make a ratio of it : p d = 3
Geometry Archimede tried to calculate π with Geometry. Here is how we can do : 1. Let’s make an equilateral triangle. Every side mesure 1cm. 2. Then, we make a hexagone with six triangles. We can say that our hexagone is almost a circle. 3. We calculate the perimeter of the hexagon (p = 6 ∗ 1 = 6cm) and the longest diagonal (d = 2 ∗ 1 = 2cm). Then, we make a ratio of it : p d = 3
Geometry Archimede tried to calculate π with Geometry. Here is how we can do : 1. Let’s make an equilateral triangle. Every side mesure 1cm. 2. Then, we make a hexagone with six triangles. We can say that our hexagone is almost a circle. 3. We calculate the perimeter of the hexagon (p = 6 ∗ 1 = 6cm) and the longest diagonal (d = 2 ∗ 1 = 2cm). Then, we make a ratio of it : p d = 3
can try some algorithms with Java. For example, this is 1’000’000 iterations of Leibniz formula : double pi=0 ; for (int n=0 ;n¡=1000000 ;n++) pi+=(Math.pow(-1,n)/(2*n+1)) ; pi*=4 ; We find 3.14159. With 100’000’000 iterations, we find 3.14159265.